A Decomposition Theory for Cyclotomic Modules under the Complete Point of View

In 1986, D. Blessenohl and K. Johnsen (1986, J. Algebra103, 141–159) proved that for any finite extension E/F of Galois fields there exists a complete normal basis generator w of E/F, which means that w simultaneously generates a normal basis for E over every intermediate field of E/F. In a recent monograph by the author (1997, “Finite Fields: Normal Bases and Completely Free Elements,” Kluwer Academic, Boston) a theory is developed which allows the study of module structures of Galois fields as extensions with respect to various subfields and which led to an exploration of the structure of complete normal basis generators as well as explicit and algorithmic constructions of these objects. In the present paper we continue the development of that theory by providing various structural results: the Complete Decomposition Theorem, the Complete Product Theorem, a Theorem on Simultaneous Generators, and a Uniqueness Theorem.